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Theorem nlmngp2 21358
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
nlmnrg.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
nlmngp2  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )

Proof of Theorem nlmngp2
StepHypRef Expression
1 nlmnrg.1 . . 3  |-  F  =  (Scalar `  W )
21nlmnrg 21357 . 2  |-  ( W  e. NrmMod  ->  F  e. NrmRing )
3 nrgngp 21340 . 2  |-  ( F  e. NrmRing  ->  F  e. NrmGrp )
42, 3syl 16 1  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   ` cfv 5570  Scalarcsca 14790  NrmGrpcngp 21267  NrmRingcnrg 21269  NrmModcnlm 21270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-nrg 21275  df-nlm 21276
This theorem is referenced by:  nlmdsdir  21360  nlmmul0or  21361  nlmvscnlem2  21363  nlmvscnlem1  21364  nlmvscn  21365
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